A craft vendor must sell at least $300 worth of merchandise to make a profit. Scarves sell for $10 each and hats sell for $20 each. She expects to sell at least 20 items. The lines on the graph represent the situation in which she makes a profit.On a coordinate plane, 2 straight lines are shown. The x-axis shows number of scarves sold, from 0 to 20, and the y-axis shows number of hats sold, from 0 to 20. The first solid line has a negative slope and goes through (2, 14) and (14, 8). Everything above the line is shaded. The second solid line has a negative slope and goes through (0, 20) and (20, 0). Everything to the right of the line is shaded.In which section of the graph does the actual solution lie?

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Answer 1

To determine which section of the graph contains the actual solution, we need to analyze the given information and the inequalities represented by the lines.

1. First line (profit line): The equation can be represented based on the information that scarves sell for $10 and hats for $20, combined with the requirement to sell at least $300:

   • \(10x + 20y \geq 300\)
   • This line is represented on the graph, with shading above it.

2. Second line (item limit line): The vendor expects to sell at least 20 items:

   • \(x + y \geq 20\)
   • This line is also represented on the graph, with shading to the right of it.

We can summarize the inequalities as:

• For the profit: \(10x + 20y \geq 300\)
• For the number of items: \(x + y \geq 20\)

Now let's break down the sections:

1. Section 1: Below the first line and below the second line.
2. Section 2: Above the first line and below the second line.
3. Section 3: Below the first line and above the second line.
4. Section 4: Above both lines.

Considering that she needs to fulfill both conditions (making a profit and selling at least 20 items), the actual solution must lie in Section 4, where both conditions (profit and item count) are satisfied simultaneously.

Therefore, the actual solution lies in Section 4.

Answer 2

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Anna is no more than 3 years older than 2 times Jamie’s age. Jamie is at least 14 and Anna is at most 35. Which system of linear inequalities can be used to find the possible ages of Anna, a, and Jamie, j?

a ≥ 3 + 2j; j ≥ 14, a ≤ 35
a ≤ 3 + 2j; j ≥ 14, a ≤ 35
a ≥ 3 + 2j; j ≤ 14, a ≤ 35
a ≤ 3 + 2j; j ≤ 14, a ≤ 35

Answer 3

To set up the inequalities based on the information given in the problem regarding Anna's and Jamie's ages, let's break it down:

1. Anna is no more than 3 years older than 2 times Jamie’s age:

   • This can be represented as: \[ a \leq 2j + 3 \]

2. Jamie is at least 14:

   • This can be represented as: \[ j \geq 14 \]

3. Anna is at most 35:

   • This can be represented as: \[ a \leq 35 \]

Now, we combine these inequalities:

• \( a \leq 2j + 3 \) (Anna's age inequality)
• \( j \geq 14 \) (Jamie's age inequality)
• \( a \leq 35 \) (Another restriction on Anna's age)

Now we can compare our inequalities with the options given:

1. \( a \geq 3 + 2j; j \geq 14, a \leq 35 \)
2. \( a \leq 3 + 2j; j \geq 14, a \leq 35 \)
3. \( a \geq 3 + 2j; j \leq 14, a \leq 35 \)
4. \( a \leq 3 + 2j; j \leq 14, a \leq 35 \)

From our analysis:

• The correct system of linear inequalities should be:

  \( a \leq 2j + 3; j \geq 14, a \leq 35 \)

Since none of the options are exactly as we've derived them, the closest correct answer based on given options is:

Option 2: \( a \leq 3 + 2j; j \geq 14, a \leq 35 \)

Note: The form of the equation in Option 2 is consistent with your constraints if we re-arrange accordingly (2j + 3 is equivalent to 3 + 2j). Thus, it can be correctly interpreted as \( a \leq 2j + 3\).

Answer 4

School rules permit no fewer than 2 teachers per 25 students. There are at least 245 students enrolled in the school. If x represent teachers and y represents students, which system of linear inequalities can be used to determine the possible number of teachers and students at the school?

2y ≥ 25x and y ≥ 245
2y ≤ 25x and y ≥ 245
25y ≤ 2x and y ≥ 245
25y ≥ 2x and y ≥ 245

Answer 5

To express the information given in the problem as a system of linear inequalities, let's break it down:

1. The school rules permit no fewer than 2 teachers per 25 students:

   • This means that for every 25 students, there must be at least 2 teachers. To represent this relationship mathematically, we can express it as: \[ \frac{y}{25} \leq \frac{x}{2} \]
   • This can be rearranged to: \[ 2y \leq 25x \quad \text{or equivalently} \quad 25x \geq 2y \]

2. There are at least 245 students enrolled in the school:

   • This can be represented as: \[ y \geq 245 \]

Now we can combine these two inequalities into a system:

• From the first point (teacher-student ratio): \[ 25x \geq 2y \]
• From the second point (minimum number of students): \[ y \geq 245 \]

Now, let's compare our derived inequalities with the options given:

1. \( 2y \geq 25x \) and \( y \geq 245 \)
2. \( 2y \leq 25x \) and \( y \geq 245 \)
3. \( 25y \leq 2x \) and \( y \geq 245 \)
4. \( 25y \geq 2x \) and \( y \geq 245 \)

Given our inequalities:

• The first inequality should be \( 25x \geq 2y \), which is equivalent to \( 2y \leq 25x \).

Thus, the correct system of linear inequalities can be written as:

Option 2: \( 2y \leq 25x \) and \( y \geq 245 \)

This correctly represents the relationships described in the problem.

Answer 6

As a computer technician, Andre makes $20 per hour to diagnose a problem and $25 per hour to fix a problem. He works fewer than 10 hours per week, but wants to make at least $200 per week. The inequalities 20x + 25y ≥ 200 and
x + y < 10 represent the situation. Which is true of the graph of the solution set? Check all that apply.

The line 20x + 25y ≥ 200 has a positive slope and a negative y-intercept.
The line x + y < 10 has a negative slope and a positive y-intercept.
The line representing 20x + 25y ≥ 200 is solid and the graph is shaded above the line.
The line representing x + y < 10 is dashed and the graph is shaded above the line.
The overlapping region contains the point (4, 5).

Answer 7

Let's analyze the inequalities given for Andre's work situation and evaluate the statements about the graph of the solution set.

1. Inequality 1: \(20x + 25y \geq 200\)

   • To find the slope and y-intercept, let's convert this to slope-intercept form (y = mx + b). \[ 25y \geq -20x + 200 \ y \geq -\frac{20}{25}x + \frac{200}{25} \ y \geq -\frac{4}{5}x + 8 \]
   • This line has a negative slope of \(-\frac{4}{5}\) and a positive y-intercept of 8.
   • Since this is a "greater than or equal to" inequality, it will be a solid line and the graph will be shaded above the line.

2. Inequality 2: \(x + y < 10\)

   • We can also convert this into slope-intercept form: \[ y < -x + 10 \]
   • This line has a negative slope of \(-1\) and a positive y-intercept of 10.
   • Since this is a "less than" inequality, it will be a dashed line and the graph will be shaded below the line.

Now, let's evaluate the truth of each statement:

1. The line \(20x + 25y ≥ 200\) has a positive slope and a negative y-intercept.

   • False. It has a negative slope and a positive y-intercept.

2. The line \(x + y < 10\) has a negative slope and a positive y-intercept.

   • True. It has a negative slope and a positive y-intercept.

3. The line representing \(20x + 25y ≥ 200\) is solid and the graph is shaded above the line.

   • True. It is a solid line since it includes equality, and the shading is above the line.

4. The line representing \(x + y < 10\) is dashed and the graph is shaded above the line.

   • False. The line is dashed, but the graph is shaded below the line.

5. The overlapping region contains the point (4, 5).

   • To check if the point (4, 5) is in the solution set:
     1. For \(20x + 25y \geq 200\): \[ 20(4) + 25(5) = 80 + 125 = 205 \quad (\text{True}) \]
     2. For \(x + y < 10\): \[ 4 + 5 = 9 \quad (\text{True}) \]
   • True. The point (4, 5) satisfies both inequalities.

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In summary, the true statements are:

• The line \(x + y < 10\) has a negative slope and a positive y-intercept.
• The line representing \(20x + 25y ≥ 200\) is solid and the graph is shaded above the line.
• The overlapping region contains the point (4, 5).

Answer 8

Toby is buying apples. Red delicious apples, x, cost $2 per pound. Braeburn apples, y, cost $3 per pound. He wants to buy at least 10 pounds of apples, but spend no more than $24. Which graph represents this situation? 1. On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (0, 8) and (9, 2). Everything below the line is shaded. The second line has a negative slope and goes through (0, 10) and (10, 0). Everything to the right of the line is shaded. 2. On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (0, 8) and (9, 2). Everything above the line is shaded. The second line has a negative slope and goes through (0, 10) and (10, 0). Everything to the left of the line is shaded. 3. On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (0, 10) and (10, 0). Everything above the line is shaded. The second line has a negative slope and goes through (12, 16) and (18, 12). Everything above the line is shaded. 4. On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (0, 10) and (10, 0). Everything above the line is shaded. The second line has a negative slope and goes through (12, 16) and (18, 12). Everything below the line is shaded.

Answer 9

To solve this problem, we need to create a system of inequalities based on the information provided about Toby's apple purchase.

Given Information:

1. Red delicious apples, \(x\), cost $2 per pound.
2. Braeburn apples, \(y\), cost $3 per pound.
3. Toby wants to buy at least 10 pounds of apples.
4. Toby wants to spend no more than $24.

Setting Up the Inequalities

1. Weight Constraint: Toby wants to buy at least 10 pounds of apples. This gives us the inequality: \[ x + y \geq 10 \]

   • This line will have a negative slope and will be shaded above (since we are interested in values that are at least 10 pounds).

2. Cost Constraint: The total cost of apples must be no more than $24. This gives us the inequality: \[ 2x + 3y \leq 24 \]

   • To express this in slope-intercept form, we rearrange it: \[ 3y \leq -2x + 24 \ y \leq -\frac{2}{3}x + 8 \]
   • This line also has a negative slope and will be shaded below (since we want to keep the total cost at or below $24).

Evaluating the Options

Now we evaluate the provided options based on the information above.

1. Option 1: First line goes through (0, 8) and (9, 2); shaded below. Second line goes through (0, 10) and (10, 0); shaded right.

   • Incorrect: The direction of shading and the points do not match the inequalities.

2. Option 2: First line goes through (0, 8) and (9, 2); shaded above. Second line goes through (0, 10) and (10, 0); shaded left.

   • Incorrect: The direction of shading does not match the inequalities; we need shading above for the weight constraint and below for the cost constraint.

3. Option 3: First line goes through (0, 10) and (10, 0); shaded above. Second line goes through (12, 16) and (18, 12); shaded above.

   • Incorrect: The second line does not match the cost constraint. Both lines shaded above also do not represent the constraints accurately.

4. Option 4: First line goes through (0, 10) and (10, 0); shaded above. Second line goes through (12, 16) and (18, 12); shaded below.

   • Incorrect: The first line is correct (shaded above), but the second line does not accurately represent the cost constraint.

Correct Identification:

From the interpretation of the inequalities:

• The correct graph would have a line for \(x + y \geq 10\) (shaded above) and a line for \(2x + 3y \leq 24\) (shaded below), which none of the provided options accurately represent.

Conclusion:

None of the provided options accurately reflect Toby's situation as described by the inequalities for his purchases. The slopes and shading directions presented in the options do not satisfy both the cost and weight constraints correctly.

Answer 10

Option 1: First line goes through (0, 8) and (9, 2); shaded below. Second line goes through (0, 10) and (10, 0); shaded right.

there is two graphs where the lines connect the top is blue then middle white and bottom red

On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (0, 42) and (42, 0). Everything to the right of the line is shaded. The second line has a negative slope and goes through (25, 35) and (32, 0). Everything to the left of the line is shaded.A collection of quarters and nickels contains at least 42 coins and is worth at most $8.00. If the collection contains 25 quarters, how many nickels can be in the collection?Let x = the number of quarters.Let y = the number of nickels.There are at least___ nickels, but no more than___ nickels.

Answer 11

Let’s break down the information given regarding the collection of coins into mathematical inequalities, and then we’ll solve the problem based on the provided conditions.

Given Information:

1. Total coins: The collection contains at least 42 coins (quarters + nickels).
2. Value of coins: The total value of the coins must be no more than $8.00.
3. Number of quarters: The collection contains 25 quarters.

Let:

• \(x\) = number of quarters
• \(y\) = number of nickels

Given that \(x = 25\):

Setting Up the Inequalities

1. Total coins inequality: \[ x + y \geq 42 \] Substituting \(x = 25\): \[ 25 + y \geq 42 \quad \Rightarrow \quad y \geq 42 - 25 \quad \Rightarrow \quad y \geq 17 \]

2. Value of coins inequality:

   • The value of the quarters is \(0.25x\) and the value of the nickels is \(0.05y\): \[ 0.25x + 0.05y \leq 8.00 \] Substituting \(x = 25\): \[ 0.25(25) + 0.05y \leq 8.00 \ 6.25 + 0.05y \leq 8.00 \ 0.05y \leq 8.00 - 6.25 \ 0.05y \leq 1.75 \quad \Rightarrow \quad y \leq \frac{1.75}{0.05} \ y \leq 35 \]

Solution

From the inequalities derived above:

• The number of nickels must be at least 17: \[ y \geq 17 \]

• The number of nickels must be no more than 35: \[ y \leq 35 \]

Conclusion

Thus, the collection can contain:

At least 17 nickels, but no more than 35 nickels.